6 (6-inch)sub rolls, slit partially open lengthwise
Combine oil, garlic, basil, oregano, lemon zest, salt, pepper, and pepper flakes in a large bowl. Transfer 2 tablespoons oil mixture to separate bowl and whisk in mayonnaise, vinegar, and lemon juice; refrigerate. Sauce can be refrigerated, covered for 2 days.
Prick chicken breasts all over with fork, cut into 1 1/4 inch chunks, and transfer to bowl with remaining oil mixture. Refrigerate, covered, for 30 minutes or up to 3 hours.
Remove chicken from marinade and thread onto skewers, about 5 pieces per skewer. You can use either metal or wood skewers, but if you are using wood, soak them in water for 15-20 minutes before using on an outdoor grill to avoid burning. I used wood skewers on my indoor grill pan without soaking and it worked just fine.
Grill chicken until browned and cooked through, 10 to 15 minutes, turning once or twice. You can use an outdoor grill (either gas or charcoal), your oven broiler or an indoor grill pan to do the job.
Transfer the chicken to sub rolls or bread, remove skewers, and drizzle with mayonnaise mixture.